In-situ observations of compressible turbulence in planetary magnetosheaths and solar wind

(1)

HAL Id: tel-01425307

https://tel.archives-ouvertes.fr/tel-01425307

Submitted on 3 Jan 2017

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

planetary magnetosheaths and solar wind

Lina Hadid

To cite this version:

(2)

NNT: 2016SACLS255

Université Paris-Saclay

École Doctorale d’Astronomie et d’Astrophisique

d’Île-de-France (ED127)

THÈSE DE DOCTORAT

Spécialté: Physique des Plasmas

Soutenue par

Lina

Hadid

Observations in-situ de la turbulence

compressible dans les magnétogaines

planétaires et le vent solaire

Directeurs:

Fouad Sahraoui et Patrick Canu

Préparée au

Laboratoratoire de Physique des Plasmas (LPP), École

Polytechnique

soutenue le 20 Septembre, 2016 Jury:

Président : M. Frédéric Moisy - Professeur (Université Paris-Sud - FAST) Reviewers : _{M. Thierry Passot} _- _{Directeur de recherche (Observatoire de Nice)}

M. William Matthaeus - Professeur (University of Delaware)

Advisor : M. Fouad Sahraoui - Directeur de recherche (Ecole Polytechnique - LPP) Examinators : Mme Karine Issautier - Directeur de recherche (Observatoire de Meudon - LESIA)

M. Vincent Genot - Astronome (Université Paul Sabatier - IRAP) Invited : _{M. Patrick Canu} _- _{Directeur de recherche (Ecole Polytechnique - LPP)}

(3)

(4)

University of Paris-Saclay

Doctoral school of Astronomy & Astrophysics of

Paris Area

P H D T H E S I S

Specialty: Plasma Physics

Defended by

Lina

Hadid

In-situ observations of compressible

turbulence in planetary magnetosheaths

and the solar wind

Advisors:

Fouad Sahraoui & Patrick Canu

Prepared at the

Laboratory of Plasma Physics (LPP), Ecole

Polytechnique

defended on the 20th _{of September, 2016}

Jury:

President : _{M. Frédéric Moisy} _- _{Professeur (Université Paris-Sud - FAST)} Reviewers : _{M. Thierry Passot} _- _{Directeur de recherche (Observatoire de Nice)}

M. William Matthaeus - Professeur (University of Delaware)

Advisor : M. Fouad Sahraoui - Directeur de recherche (Ecole Polytechnique - LPP) Examinators : _{Mme Karine Issautier} _- _{Directeur de recherche (Observatoire de Meudon - LESIA)}

M. Vincent Genot - Astronome (Université Paul Sabatier - IRAP) Invited : M. Patrick Canu - Directeur de recherche (Ecole Polytechnique - LPP)

(5)

(6)

(7)

(8)

(9)

(10)

Abstract

Among the different astrophysical plasmas, the solar wind and the planetary magnetosheaths represent the best laboratories for studying the properties of fully developed plasma turbulence. Because of the relatively weak density fluctuations (∼ 10%) in the solar wind, the low frequency fluctuations are usually described using the incompressible MHD theory. Nevertheless, the effect of the compressibility (in particular in the fast wind) has been a subject of active research within the space physics community over the last three decades.

My thesis is essentially dedicated to the study of compressible turbulence in different plasma en-vironments, the planetary magnetosheaths (of Saturn and Earth) and the fast and slow solar wind. This was done using in-situ spacecraft data from the Cassini, Cluster and THEMIS/ARTEMIS satellites.

I first investigated the properties of MHD and kinetic scale turbulence in the magnetosheath of Saturn using Cassini data at the MHD scales and compared them to known features of the solar wind turbulence. This work was completed with a more detailed analysis performed in the magnetosheath of Earth using the Cluster data. Then, by applying the recently derived exact law of compressible isothermal MHD turbulence to the in-situ observations from THEMIS and CLUSTER spacecrafts, a detailed study regarding the effect of the compressibility on the energy cascade (dissipation) rate in the fast and the slow wind is presented. Several new empirical laws are obtained, which include the power-law scaling of the energy cascade rate as function of the turbulent Mach number. Eventually, an application of this exact model to a more compressible medium, the magnetosheath of Earth, using the Cluster data provides the first estimation of the energy dissipation rate in the magnetosheath, which is found to be up to two orders of magnitude higher than that observed in the solar wind.

Keywords: Turbulence, compressible turbulence, astrophysical plasmas, solar wind, planetary magnetosheaths, Earth, Saturn, in-situ observations

Résumé

Parmi les différents plasmas spatiaux, le vent solaire et les magnétogaines planétaires représen-tent les meilleurs laboratoires pour l’étude des propriétés de la turbulence. Les fluctuations de den-sité dans le vent solaire étant faibles, à basses fréquences ces dernières sont généralement décrites par la théorie de la MHD incompressible. Malgré son incompressibilité, l’effet de la compressibilité dans le vent solaire a fait l’objet de nombreux travaux depuis des décennies, à la fois théoriques, numériques et observationnels.

Le but de ma thèse est d’étudier le rôle de la compressibilité dans les magnétogaines plané-taires (de la Terre et de Saturne) en comparaison avec un milieu beaucoup plus étudié et moins compressible (quasi incompressible), le vent solaire. Ce travail a été realisé en utilisant des données in-situ de trois sondes spatiales, Cassini, Cluster et THEMIS B/ARTEMIS P1.

La première partie de mon travail a été consacrée à l’étude des propriétés de la turbulence dans la magnétogaine de Saturne aux échelles MHD et sub-ionique, en comparaison avec celle de la Terre en utilisant les données Cassini et Cluster respectivement. Ensuite j’ai appliqué la loi exacte de la turbulence isotherme et compressible dans le vent rapide et lent en utilisant les don-nées THEMIS B/ARTEMIS P1, afin d’étudier l’effet et le rôle de la compressibilité sur le taux de transfert de l’énergie dans la zone inertielle. Enfin, une première application de ce modèle dans la magnétogaine de la Terre est présentée en utilisant les données Cluster.

(11)

(12)

iii

Acknowledgments

My experience at the Laboratory of Plasma Physics (LPP) has been nothing short of amazing. This thesis represents not only my work at the keyboard, it is a milestone in 3 years of work within the space plasma group. My research has been also the result of many experiences I have encountered in the lab and on the campus from dozens of remarkable individuals who I also wish to acknowledge.

At first, I could not have finished my work without the help and patience of my dear supervisor Dr. Fouad Sahraoui who gave me the opportunity to work with him even before my PhD, since my master internship and has been supportive since then. Fouad, I would like to express my sincere gratitude, for your confidence, patience and encouragements. Your scientific insights, advices, and remarks have always guided me to progress in my work, not to forget the scientific arabic that you taught me!

I would like to thank as well to Dr. Patrick Canu for his fruitful comments on the use of the Cassini data and on this manuscript. Prof. Sebastien Galtier and Dr. Supratik Banerjee with whom I got to work in the last part of my thesis and I had the chance to "see" the turbulence problem from a theoretician point of view, a different angle than a data analysist. Dr. Shiyong Huang for the fruitful collaboration on analyzing the Cluster data. Dr. Ronan Modolo who enlighted me regarding the pointing problem of the CAPS on Cassini and gave me his code to recompute the ion velocity in the field-of-view of the intsrument. I have to mention and thank as well Dr. Thierry Passot for the interesting discussions we had, Dr. Vincent Génot and Dr. Adam Masters for providing me the Cassini data events that I used during my work, and for their valuable scientific advices.

I must recognize that this work would have not been possible without the sup-port of the the doctoral school of Astronomy and Astrophysics (ED127), and the University of Paris-Sud.

I would like to thank as well all the members of the jury for their participation and for evaluating this work, in particular the referees for their interest in carefully reading and commenting my manuscript.

(13)

A special thanks to Dr. Khurom Kiyani, with whom I had the chance to share my office and profit from his deep knowledge and special lessons about data analysis, plasma turbulence but also life. To Patrick Robert, who has been a main source of invaluable motivation and support since I started working at LPP. Alexis Jean-det who introduced me to Python and shared his unique and massive knowledge in programming, I thank him as well for his great and inspiring suggestions and comments.

Part of the data analysis was done with the AMDA science analysis system provided by the Centre de Données de la Physique des Plasmas (CDPP) supported by the CNRS and the CNES. So I thank them for this practical platform, that helped me choosing and ploting the data.

A warm thanks also for my friends and colleagues in the lab: Sergey Stepanyan, Sergey Sherbanev, Alexandros Chasapis, Claudia Rossi, Sumire Kobayashi, Ma-lik Mansour, Ralph Katra, Alexis Jeandet, Pierre Morel, Pascaline Grondin, Katy Ghantous, Christelle Barakat, and the great evennings we spent "Chez Richard". I would also like to humbly thank the various undergraduate students of the physics dept who, through the course of tutoring them, helped me learn some of the physics I should have learnt when I was an undergraduate.

Not to forget of course Eva, Anais and Iyad, with whom I shared the flat for almost three years, during all my PhD period. I thank you "froomies" for the many amazing memories I have now with you and for all the great and priceless times we lived and spent together!!! My close lebanese friends, Iman, Hiba, Julie, Violette, Angy, Nidale, I thank you so much for your daily encouragements, and for making me feel home especially with the great food you prepared!

I would like also to thank my running team (Michael Molle, Catherine Sarazin, Vanessa Moreau), who motivated me to keep on running and exercising, in order to de-stress during the toughest period of writing my thesis.

(14)

v

Abbreviations

ACE Advanced Composition Explorer AIC Alfvén Ion Cyclotron

AMDA Automated Multi Dataset Analysis

AMPTE Active Magnetospheric Particle Tracer Explorers

ARTEMIS Acceleration Reconnection Turbulence and Electrodynamics of the Moon’s Interaction with the Sun

ASI Agenzia Spaziale Italiana AU Astronomical Unit

BG13 Banerjee & Galtier 2013 C09 Carbone et al. 2009

CAPS Cassini Plasma Spectrometer

CETP Centre d’Etude des Environnements Terrestre et Planétaires CIR Corotating Interaction Region

CIS Cluster Ion Spectrometer

CNES Centre National d’Etudes Spatiales

CNRS Centre National de la Recherche Scientifique CODIF Composition Distribution Function

CSA Cluster Science Archive ED Ecole Doctorale

EFI Electric Field Instrument EFW Electric Field and Waves ELS Electron Spectrometer EM Engineering Model

EMIC Electromagnetic Ion Cyclotron ESA European Space Agency

ESAs Electrostatic analyser

(15)

FFT Fast Fourrier Transform FGM Fluxgate Magnetometer FM Flight Model

GMC Giant Molecular Clouds GPS Global Positioning System GSE Geocentric Solar Ecliptic

GSM Geocentric Solar Magnetospheric HBR High Bit Rate

HIA Hot Ion Analyser

IAS Institut d’Astrophysique Spatiale IBS Ion Beam Spectrometer

IDL Interactive Data Language

IDPU Instrument Data Processing Unit IK Iroshnikov-Kraichnan

IMS Ion Mass Spectrometer

IRAP Institut de Recherche en Astrophysique et Planétologie IRF Institutet for RymdFysik

IRM Ion Release Module ISM Interstellar Medium JPL Jet Propulsion Laboratory KAW Kinetic Alfvén Waves

KRTP Kronocentric Coordinate system KSM Kronocentric Solar Magnetospheric

LATMOS Laboratoire Atmosphères Milieux Observations Spatiales

LESIA Laboratoire d’Etudes Spatiales et d’Instrumentation en Astrophyisque LFR Low Frequency Receiver

(16)

vii MAG Magnetometer

MFA Mean Field Aligned

MFR Medium Frequency Receiver MHD Magnetohydrodynamics MMS Magnetospheric Multiscale MS Magnetosheath

MSSL Mullard Space Science Laboratory

NASA National Aeronautics and Space Administration NBR Normal Bit Rate

NEMI Noise Equivalent Magnetic Induction NI Nearly Incompressible

PDF Probability Density Function PDS Planetary Data Science

PEACE Plasma Electron and Current Experiment PIC Particle In Cell

PP98 Politano and Pouquet 1998 PSD Power Spectral density

RPWS Radio and Plasma Waves Science SA Spectrum Analyser

SCM Search-Coils Magnetometer SNG Single

SST Solid State Telescope

STAFF Spatio-Temporal Analysis of Field Fluctuations SW Solar Wind

THEMIS Time History of Events and Macroscale Interactions during Substorms ToF Time of Flight

(17)

ULF Ultra Low Frequency VAP Van Allen Probes

VDF Velocity Distribution Function WFR Waveform Receiver

WHAMP Waves in Homogeneous Anisotropic Magnetized Plasma

(18)

List of Figures

1.1 From left to right: turbulent motions in thin film of soapy water, in coffee, in clouds, in the atmosphere, in the Sun and in the galaxy. . . 4

1.2 A solution in the Lorenz attractor. . . 5

1.3 (a) Fast and slow solar wind resulting in compression zone because of the solar rotation (adapted from Pizzo [1978]), (b) The different solar wind speed and the magnetic field polarity observations from the Ulysses spacecraft (adapted from McComas et al. [1998]). . . 7

1.4 Dependence of the proton (a) parallel and (b) perpendicular temper-atures on the radial distance from the sun. Adapted from Marsch et al. [1982]. . . 8

1.5 Schematic of the terrestrial magnetosphere showing the distended field lines of both day and night side magnetospheres. RM P denotes

the distance to sub-solar magnetopause. c_{Fran Bagenal & Steve} Bartlett. . . 9

1.6 Schematic representation of a quasi-parallel and a quasi-perpendicular bow-shock. . . 10

2.1 Transition from a (a) laminar to a (d) turbulent regime as function of the Reynolds number Re for a flow passing a cylindrical obstacle.

a) Re= 1.54, b) Re= 9.6, c) Re = 13.1, d) Re = 26. Adapted from

Frisch [1995]. . . 16

2.2 A schematic showing the importance of the higher-order statistics. The higher is the moment, the more information it gives regarding the few bursty events in the tails of the PDFs. . . 19

2.3 Illustration of the Richardson-Kolmogorov cascade, in the real space (a) and Fourier space (b). . . 20

2.4 (a) Illustration of the Iroshnikov-Kraichnan phenomenology of in-compressible MHD turbulence and (b) the corresponding image à la Richardson-Kolmogorov in Fourier space showing a typical interplan-etary magnetic field power spectrum at 1 AU. Adapted respectively from [Supratik, 2014] and [Bruno & Carbone, 2005]. . . 22

3.1 Typical power spectral density of the magnetic field fluctuations in the ecliptic solar wind at 1 AU, combining ACE and Cluster data. The vertical dashed lines indicate the correlation length (λc), the

ion gyro-radius (ρi) and the electron gyro-radius ρe. Adapted from

(23)

3.2 Left: Normalized PDF of the velocity field fluctuations at different scales τ using Helios 2 data in the fast solar wind. Solid lines represent the fit made using a log-normal model [Sorriso-Valvo et al., 1999]. Right: The scaling exponent as a function of the different orders p computed in the solar wind at ∼ 9 A.U. using Voyager data [Burlaga, 1991]. . . 33

3.3 Nearly sinusoidal mirror mode evidenced by the strong anti-correlation of the magnetic field (solid) and ion density (dashed) measurements. Adapted from Leckband et al. [1995]. . . 34

3.4 "Star formation and magnetic turbulence in the Orion Molecular Cloud". The texture represents the magnetic field’s orientation. Blue color correspond to regions with low dust density, while the yellow and red areas reflect denser (and mostly hotter) clouds. c_{ESA and} the Planck Collaboration. . . 37

3.5 Fourier spectra of the density weighted velocity. Adapted from Fed-errath et al. [2010]. . . 38

3.6 Volume-weighted density PDFs p(s) of the logarithmic density s in linear scaling. Adapted from Federrath et al. [2010]. . . 39

3.7 Comparison between the mixed third-order compressible pseudo-energy flux w±_{(red) with the incompressible flux (green) as computed using}

Ulysses data. Adapted from Carbone et al. [2009]. . . 41

4.1 Artistic photo of Cassini-Huygens during the Saturn Orbit Insertion (SOI) maneuver. c_NASA/JPL.. . . 46

4.2 The Cassini-Huygens spacecraft assembly in Kennedy Space Center’s. c

NASA Kennedy Space Center. . . 47

4.3 Left: The location of the magnetometer hardware on the spacecraft is shown halfway the "mag boom". Right: Electronics photograph of the FGM (with cover off) and electronics board. Adapted from Dougherty et al. [2002]. . . 49

4.4 Left: The placement of the different RPWS hardware on Cassini Orbiter. Right: The tri-axial magnetic search coils assembly covered by thermal blankets. Adapted from Gurnett et al. [2004]. . . 49

4.5 The functional block diagram of the RPWS instrument. Adapted from Gurnett et al. [2004]. . . 50

4.6 Left: Location of the CAPS sensors on Cassini spacecraft. Right: CAPS instrument with its three sensors (ELS, IMS, & IBS) and the actuator motor platform. Adapted from Young et al. [2004]. . . 51

4.7 Cluster mission with its 4 spacecrafts orbiting around Earth. c_ESA. 52

4.8 Left: FGM instrument c_{Imperial College. Right: STAFF covered} by the thermal blanket and the preamplifier c_LPP. . . . 53

(24)

List of Figures xv

4.10 a) 4 boxes of boom units and electronic box for EFW experiment. Each boom unit contains spherical probe and 50 m of wire boom. b) EFW probe configuration c_IRF. . . . 54

4.11 a) Density (ne) as a function of the spacecraft potential (Vsc) relative

to the solar wind. The best fit exponential calibration curve is shown as a solid purple line. b) Comparison between the high frequency den-sity inferred from the Vsc (black) and the lower frequencies measured

by PEACE at the spin (4 s) resolution (red). . . 55

4.12 (a-c) Magnetic field modulus and (b-d) electron plasma density mea-sured by the Cassini spacecraft in the solar wind (SW) and in the magnetosheath (MS) of Saturn on 2005 March 17 from 00:00 to 10:00 and 2007 February 03 from 00:10 to 02:00 for a quasi-perpendicular and quasi-parallel shocks respectively. . . 57

4.13 Power spectral density of the magnetic field fluctuations (δB) mea-sured between 02:00 and 08:30. The black lines are the power-law fits. The arrow corresponds to the ion gyrofrequency fci; the gray and the

red shaded bands indicate the Taylor-shifted ion inertial length (fdi)

and Larmor radius (fρi), respectively. The width reflects the

uncer-tainty due to errors in estimating the ion moments (see caveats in Section 4.4.1). Adapted from [Hadid et al., 2015]. . . 58

4.14 Histograms of the spectral slopes at MHD and sub-ion (kinetic) scales downstream of (a) quasi-perpendicular and (b) quasi-parallel shocks of Saturn’s magnetosheath using Cassini data. Adapted from [Hadid et al., 2015]. . . 59

4.15 Top: Histogram of the spectral slope distribution for the 400 time intervals at the MHD scales (a) and the sub-ion ones (b). Bottom: The corresponding 2D distribution (XKSM, YKSM ) of the spectral

slopes at MHD scales (c), and the sub-ion scales (d). The black curves represent the bow-shock and the magnetopause positions computed using the semi-empirical model of Went et al. [2011] and the empirical magnetopause model of Kanani et al. [2010] respectively. . . 61

4.16 Top: Power spectral density of δB showing a positive/flat power law below 0.01 Hz. The red lines are the power-law fits. The green curve corresponds to the estimated noise level of FGM. Bottom: The local slope variation with respect to the frequency. The red curve is a fit using a hyperbolic tangent function and the dashed line correspond to the frequency break at the ion gyro-radius. . . 62

(25)

4.18 Top: Histograms of slope values (a) at MHD scales and (b) the sub-ion ones. Bottom: The corresponding 2D distributsub-ion Pα(XGSE, YGSE)

of the spectral slopes (c) at MHD and the (d) the sub-ion scales. The blue curves represent the average magnetopause and bow shock po-sitions computed using the paraboloidal bow-shock model of Filbert & Kellogg [1979] and the magnetopause model of Sibeck et al. [1991]. The numbers represent the total number of the case studies. . . 64

4.19 PDFs of the magnetic field increments in the energy containing and sub-ion scales (blue and red respectively) downstream of quasi-perpendicular (a-b) and quasi-parallel (c) shocks (with Poisonnian error bars). Nor-malized histograms with 300 bins each were used to compute the PDFs. The same values of τ were used in both cases, τ ∼ 12 s for the kinetic scales and τ ∼ 470 s for the MHD scales. All the PDFs have been rescaled to have a unit standard deviation. A Gaussian distribution (black dashed curve) is shown for comparison. Adapted from [Hadid et al., 2015].. . . 65

4.20 Kurtosis (fourth standardized moment) of magnetic field increments as a function of the separation length s. Adapted from [Karimabadi et al., 2014]. . . 66

4.21 Different orders of the structure functions of the magnetic field in-crements δBτ(t) as function of the time lag τ downstream of

quasi-perpendicular (a-b) and quasi-parallel (c) shocks. (d), (e) and (f) represent the corresponding scaling exponent ζ(m). Adapted from [Hadid et al., 2015]. . . 66

4.22 Comparison between the theoretical magnetic compressibilities, com-puted from the linear solutions of the compressible Hall-MHD (color dotted line) and of the Vlasov-Maxwell equations (colored solid line) for β = 1 and ΘkB = 87◦, with the observed one from the data of

Fig. 4.12 (02:00-08:30) (solid black curve). The Taylor hypothesis was used to convert the frequencies in the spacecraft frame into wavenum-ber. The red, green and blue curves correspond respectively to the theoretical fast, slow and KAW modes. The horizontal dashed black line at CB = 1/3 indicates the power isotropy level. Adapted from

[Hadid et al., 2015]. . . 69

4.23 The theoretical magnetic compressibilities from the linear solutions of the compressible Hall-MHD, for the Alfvén, fast and the slow modes. The first column represent the solutions by fixing ΘVB = 87◦ and

varying β from 0.2 to 90, whereas for the second column, β was fixed to ∼ 1 and ΘVB varying between a parallel to a

quasi-perpendicular propagation. . . 70

(26)

List of Figures xvii

4.25 (a-b) Magnetic field magnitude and plasma density measured by the FGM and CIS and PEACE experitments onboard Cluster 3, (c-d) the corresponding FFT spectra of the magnetic fluctuations, (e-f) the global (black) and local (green) magnetic compressibilities CB. . 72

4.26 (a-b-c) Estimated magnetic compressibilities in the magnetosheath of Earth using Cluster 3 spacecraft data. The histogram of (d) shows the averaged values of CB in the noted frequency range. . . 73

4.27 Histograms of the kinetic Alfvén ratio (δ˜n)2_/(δb

⊥)2 in the solar wind

defined in 4.3. Adapted from [Chen et al., 2013]. . . 75

4.28 Spectrum of the density and magnetic fluctuations in the terrestrial magnetosheath, normalized according to Equations 4.3. The black dashed line represents the limit of when the spectra reach the noise level of SCM. Between ion and electron scales the spectra are of δ˜n ≫ δb⊥, ruling out the presence of incompressible whistler turbulence. . 76

4.29 The distribution PR(XGSE, YGSE) of the estimated ratio R = log1/τ c fci

of the turbulent fluctuations for all the events displayed in Figure 4.18-a.. . . 78

4.30 All-sky projection of the CAPS-IMS field of view (FoV). The shaded areas represent surrounding spacecraft instruments obscuring the CAPS FoV. Similar encroachments occur for IBS and ELS. Adapted from Young et al. [2004]. . . 80

4.31 Electron density measured by CAPS/ELS and estimated using the Langmuir probe measurments. Ion density measured by the IBS sensor. 81

4.32 Saturn’s Solar Ecliptic coordinate systems. . . 82

4.33 Left: SNG ion fluxes measured by CAPS plotted in the magne-tosheath of Saturn, on 2005/03/17, from 02:00 to 04:30 as a function of the latitudinal (δ) and longitudinal (θ) location of the spacecraft. Right: The corresponding 1D energy spectra of the single ion fluxes observed by each of the eight anodes of CAPS. The red curve iden-tifies the anode number 8 where the ion flux takes its highest value.

. . . 82

4.34 From Top to Bottom: The total number of counts averaged over the eight anodes (SNG) for the same case study presented earlier (2005/03/17), the corresponding velocity components (dashed line with triangles) compared with the velocity components provided by the PDS (solid line with circles). Last panel: relative differences between the modulus of the velocity provided by the PDS |V |P and

the one computed from LATMOS code (|V |L) . . . 83

4.35 Cassini PSD of the magnetic field flufctuations of the WFR calibrated at LPP (blue curve) and Iowa University (black curve), for each of the components (Bx, first panel; By, second panel; Bz, third panel).. 85

(27)

4.37 Top: Cassini engineering model (EM) sensor located within the new Helmholtz coils inside the new facility. One can see that YY’ coils are connected in the connection panel (on the bottom right side of the first image) for which the 50 Hz noise is the smallest. In the second image, one can see the EM preamplifier connected to the sensor with the EM and connected to the rack GSE2 (+12/-12V power supply). The Y axis (red coaxial cable) is connected to the spectrum analyzer and the oscilloscope (Bottom Figure). . . 86

4.38 Top: Comparison between Cassini EM (gain) in dBV/nT at Chambon-la-Forêt facility and the previous FM measurements (back in the 90’s in a different facility). Bottom: Cassini EM ground noise level (NEMI). The observed peaks correspond to the harmonics of the 50 Hz fundamental waveform. . . 87

5.1 Artist concept of the five THEMIS space spacecraft traveling through the magnetic field lines around Earth. c NASA . . . 94

5.2 Top view of the five THEMIS spacecraft in their "wedding cake" configuration before spin-balance testing. c_{NASA/George Shelton.}. 95

5.3 Flight unit of the FGM instrument (silver-colored cylinder at the foreground) and of the Search Coil Magnetometer (SCM) instrument (3 black colored, orthogonal rods at the background) mounted on their carbon-composite booms. The combination of these instruments measures the ambient magnetic field and its oscillations up to 4 kHz.

c

NASA. . . 96

5.4 First flight model of the Electrostatic Analyzer (ESA) instrument (round structure) and the Instrument Data Processing Unit (IDPU).

c

NASA. . . 97

5.5 THEMIS spacecraft instruments. c_NASA. . . . 98

5.6 The average solar wind speed and the total plasma β for all the used data intervals using THEMIS B/ARTEMIS P1 spacecraft. . . 100

5.7 In each Figure from top to bottom: the solar wind magnetic field components, ion velocity, ion number density, ΘVB angle and total

plasma beta (β = βi+ βe) measured by the FGM and ESA

experi-ments onboard the THEMIS B/ARTEMIS P1 spacecraft. . . 101

5.8 Top: comparison of the different terms |F1|, |F2| and |F3| of the flux

FC+Φin the fast and the slow wind. Bottom: comparison between the

corresponding turbulent cascade rates (ε) given by the PP98 (black) and BG13 (red) models. The compressibilities in the fast and the slow wind, defined as p(hρ2_{i − hρi}2_{)/hρi, are ∼ 13.7% and ∼ 15%}

respectively. The inserted panel, shows a few examples for which BG13 model gives a relatively more regular cascade rate over two decades, compared to the PP98 model. . . 102

(28)

List of Figures xix

5.10 Histograms of the h|εC|i estimated by BG13 (red) and h|εI|i estimated

by PP98 (blue) in the fast and slow winds.. . . 104

5.11 Histograms of the ratio between the compressible to the incompress-ible cascade rate R = h|εC|i/h|εI|i in the fast (red) and slow (blue)

winds. . . 104

5.12 Variation of the compressible cascade rate h|εC|i as a function of the

compressibility (p(hρ2_{i − hρi}2_{)/hρi) and the wind speed.}_{. . . 105}

5.13 The absolute averaged energy transfer rate as a function of the tur-bulent Mach number in the fast and the slow wind. The black line represents a least square fit of the data in the log space. α is the corresponding correlation coefficient. . . 106

5.14 The compressibility (_p(hρ2_{i − hρi}2_{)/hρi) as a function of the}

turbu-lent Mach number in the fast and the slow wind. The black line represents a least square fit of the data. α is the corresponding cor-relation coefficient. . . 106

5.15 First panel: the magnetic (blue), kinetic (red) and internal (green) energies of the compressible fluctuations in the fast and slow winds. Second panel: the total compressible energy, plotted as a function of the compressible energy cascade rate h|εC|i. The black lines are the

least-square-fits with α the corresponding fit coefficient. δE ∼ εα

C →

εC ≡ δE1/α. . . 107

5.16 Estimation of the contribution of the compressible fluxes w.r.t. in-compressible (Yaglom) flux to the in-compressible cascade rate for the fast (right) and slow (left) winds. . . 109

5.17 Estimation of the contribution of the compressible fluxes w.r.t. in-compressible (Yaglom) flux to the in-compressible cascade rate for the fast (right) and slow (left) winds as function of the ratio between the compressible to incompressible energy of the turbulent fluctuations. . 109

5.18 Histograms of the signed energy cascade rate estimated using the compressible model BG13 in the fast (left) and slow (right) solar wind. The red lines represent the corresponding mean values of the energy cascade. . . 110

5.19 The correlations between the estimated signed incompressible and compressible cascade rates εI and εI in the fast (red) and slow (blue)

winds. . . 111

5.20 The compressible cascade rate εC plotted as function of the

cross-helicity and the solar wind speed. . . 111

5.21 Cross-helicity (σc) plotted as a function of the propagation angle ΘVB

and h|εC|i in the fast wind (Top) and the slow wind (Bottom). For

an outward propagation, σc ∼ +1 and is anti-parallel to B. Whereas

for an inward propagation σc ∼ −1 and is parallel to B. . . 112

(29)

5.23 Energy dissipation rate (h|εC|i) as a function of the averaged

propa-gation angle ΘVBand the total compressible energy computed in the

fast and the slow wind.. . . 114

5.24 The energy dissipation rate computed using BG13 (red), PP98 (black), C09 (blue), C09 corrected (light blue). . . 116

5.25 Histogram of the ratio R = h|εC09|i / h|εC|i using the model C09

model (blue) and the modified one (red). . . 116

5.26 2D plot of h|F1|i/h|F3|i as a function of the compressibility and the

ratio h|εC|i/h|εC|i > 1. . . 117

5.27 Histograms of εI computed in the fast wind at different values of

τ = [21, 81]. The red line represents the average value of εI for each

value of τ . . . 119

5.28 Example of the effect of ΘVB rotation. Left: from top to bottom,

magnetic field components, the ion velocity, ion density, propagation angle, total beta. Right: comparison between h|εC|i computed on a

time interval including sharp discontinuities (cyan, 04:40-06:00), and by excluding them (dark blue 05:05-05:38). . . 120

6.1 Averaged flow velocity and the total plasma beta in the terrestrial magnetosheath. . . 124

6.2 Magnetosheaths events for which the inertial range is characterized by (left) an Alfvénic type of the magnetic compressibility and (right) a magnetosonic like one. From top to bottom: magnetic field vector, velocity vector, ion density, propagation angle ΘVB, and the total

plasma beta. . . 125

6.3 (a-b) The energy dissipation rates computed for the two particular events using the isothermal compressible MHD model (BG13), in a comparison with the incompressible model (PP98). . . 125

6.4 Histograms of the averaged values of the compressible energy cascade rate computed using BG13 (blue) compared with the incompressible one calculated using PP98 (red), for two groups of events: (a) showing and Alfvénic-like CB and (b) a magnetosonic-like CB. . . 126

6.5 Total energy cascade rate as a function of the plasma compressibility for the Alfvénic and the magnetosonic-like events. The black line is a least square fit of the measurements with α the corresponding fitting index. . . 127

6.6 Compressible energy cascade rate as a function of the turbulent Mach number for the Alfvénic and the magnetosonic like events. The black line is a least square fit of the measurements with α the corresponding fitting index. . . 127

(30)

List of Figures xxi

6.8 First Panel: the magnetic (blue), kinetic (red) and internal (green) energies of the compressible fluctuations in the terrestrial magne-tosheath. Second Panel: the total compressible energy, plotted as a function of the compressible energy cascade rate h|εC|i. The black

lines are the least-square-fits with α the corresponding fit coefficient. δE ∼ εαC → εC ≡ δE1/α. . . 129

6.9 Estimation of the contribution of the compressible fluxes w.r.t. in-compressible (Yaglom) flux to the in-compressible cascade rate for the Alfvénic (right) and the magnetosonic-like (left) events as a function of the ratio between the compressible to incompressible energy of the turbulent fluctuations. . . 130

6.10 Cross-helicity (σc) as a function of the propagation angle ΘVB for

(31)

(32)

List of Tables

1.1 Typical parameters for astrophysical plasmas. Adapted from [Schekochi-hin et al., 2009].. . . 6

1.2 Typical (approximated) plasma parameters at different radial dis-tances in the solar wind and in different planetary magnetosheaths (the density (n), background magnetic field (B0), ion Larmor radius

(ρi = Vthi/2πfci where fci = qB0/2πmi is the ion gyro-frequency),

the ion plasma (βi), the Alfvénic Mach number (MA = Vf low/VA)

and the turbulent Mach numbers (Ms=pδv2/Cs2)). . . 12

(33)

(34)

Part I

(35)

(36)

Chapter 1

Introduction

Contents

1.1 Turbulence overview . . . 3

1.2 Physical context: the solar wind and planetary magneto-spheres. . . 5

1.2.1 The solar wind . . . 6

1.2.2 Planetary magnetospheres . . . 9

1.3 Motivations and outline. . . 11

1.1 Turbulence overview

Turbulence is ubiquitous in the universe; it is almost the rule in flowing fluids. In everyday life, the turbulent motion covers a wide range of time and scales: it is observed in any disturbed hydrodynamical fluid (tap water, smoke rising from a cigarette or a fire, large-scale structure of the atmospheric circulation, etc...), in plasma laboratories (Laser- matter interactions, nuclear fusion reactors) and in astrophysical plasmas(e.g, the solar wind streaming outward from our Sun, mag-netospheric plasmas and the interstellar medium). Other than the usual hydrody-namical fluids and space plasmas, turbulence is also believed to occur in other fields like quantum mechanics [Proment et al.,2009], non-linear optics and dynam-ics[Garnier et al.,2012;Parker et al.,2015] and in vibrational mechanics [Barba & Velasco Fuentes,2008]. From the quantum scales up to the macroscopical ones, turbulence is characterized by the formation/presence of randomly moving eddies (Figure 1.1).

(37)

especially that with climate change, it is a key for understanding the heat exchange between the ocean and the atmosphere. Perhaps even market fluctuations [ Mandel-brot,1999] may benefit from a better understanding of the problem of turbulence. Yet, despite its importance, turbulence largely remains an "unsolved" problem in the sense that a clear physical understanding of the observed phenomena does not exist. Indeed we still do not understand in complete detail how or why turbulence occurs, nor can we predict turbulent behavior with any degree of reliability, even in very simple flow situations. And worse, we even disagree about what we think we know about it.

Figure 1.1: From left to right: turbulent motions in thin film of soapy water, in coffee, in clouds, in the atmosphere, in the Sun and in the galaxy.

(38)

1.2. Physical context: the solar wind and planetary magnetospheres 5 characterized by large numbers of degrees of freedom which is seen for example when numerous spatial scales are excited non-linearly. However, from the statistical point of view, it is possible to predict the turbulent behavior, the reason for which it is important to study turbulence with statistical tools, as we will see in Chapter 2.

Figure 1.2: A solution in the Lorenz attractor.

1.2 Physical context: the solar wind and planetary

mag-netospheres

(39)

2005; Retinò et al., 2007] and to compute spatial gradients of quantities such as the magnetic field B and plasma flow velocity V, yielding key quantities like the electric current J and the vorticity ω. Last but not least, the near-Earth space and the planetary magnetospheres cover a wide range of values of the plasma parameters (e.g., the plasma β = Vthi/VA, the Alfvén Mach number MA = Vf low/VA and

tur-bulent Mach number Ms =pδv2/Cs2, where Vthi =p2KTi/mi is ion the termal

speed, VA= B0/√µ0n0mi represents the Alfvén speed, Vf low, the flow velocity, δv,

the velocity fluctuations, Cs the ion sound speed. K, Ti, mi, B0, µ0, n0 are

respec-tively, the Boltzmann constant, the ion temperature, the ion mass, the background magnetic field, the vacuum permeability, and the plasma density), which allow us to extrapolate the results obtained in these regions to other astrophysical plasmas that have (or thought to have) similar values (See table 1.1).

Table 1.1: Typical parameters for astrophysical plasmas. Adapted from [Schekochihin et al.,2009].

1.2.1 The solar wind

The solar wind is a stream of ionized particles that comprises ionized hydrogen (96%), a small proportion (4%) of ionized helium and a small fraction of heavy ions like Fe, Si and O [Bame et al.,1968,1975]. It is continuously blowing out from the solar corona into the interplanetary space to terminate somewhere in the interstellar space around 120 AU (Astronomical Unit ∼ 150 × 106 _{km). It originates from the}

(40)

1.2. Physical context: the solar wind and planetary magnetospheres 7 300 km/s and 400 km/s at 1 AU. The origin of those two types of winds is still debated, however, the fast wind apparently emanates from the magnetically open coronal holes which are representative of the inactive Sun at the polar region [Hassler et al., 1999]. The slow wind on the contrary is generally ejected from lower solar latitudes, the equatorial zone of the Sun [McComas et al.,2000].

a b

Figure 1.3: (a) Fast and slow solar wind resulting in compression zone because of the solar rotation (adapted from Pizzo [1978]), (b) The different solar wind speed and the magnetic field polarity observations from the Ulysses spacecraft (adapted fromMcComas et al. [1998]).

The acceleration process of the fast wind is still not fully understood and cannot be fully explained by Parker’s theory. In fact, the latter predicts a velolcity of 500 km/s for a coronal temperature of around 106 _{K, but in reality the speed of the fast wind}

is around 700 km/s for a lower temperature at the coronal holes (∼ 105 _{K). It has}

been suggested that some acceleration processes related to the dissipation of high frequency Alfvén waves could explain the higher wind speed [Tu & Marsch, 1997]. The origin of the slow wind is also unclear. One can see in Figure 1.3(b) that the slow wind emanates from the equatorial belt which is magnetically complex. The fast and the slow winds have generally different plasma and thermodynamic properties:

1. The density in the slow wind is higher than that in the fast wind, and so its mass flow (ρv) gets even higher than the fast one.

2. The fast wind is relatively stationnary and displays slower changes with time, while the slow one is highly variable.

(41)

heliocentric distances the pressure waves bounding a CIR commonly steepen into a shock front [Gosling & Pizzo,1999].

4. Interestingly, data from Helios showed that the proton temperature parallel to the magnetic field in the slow wind drops more rapidly with the radial distance than in the fast flow [Marsch et al.,1982] (Figure1.4(a)).

a b

(42)

1.2. Physical context: the solar wind and planetary magnetospheres 9

Howes et al. [2008a]; Schekochihin et al. [2009]; Sahraoui et al. [2009, 2010]; Gary et al. [2012]) and those who do not call into play those processes and rather argue in favor of localized dissipation. This latter can occur via magnetic reconnection within the current sheets that form naturally in turbulent plasmas (Matthaeus & Goldstein [1986]; Matthaeus et al. [2005]; Osman et al. [2011a]; Karimabadi et al.

[2014]). Note however that this view is called into question by other authors who argue that wave-particles interactions can still play a role within the reconnection regions (Loureiro et al. [2013]; Howes [2015]). These questions will not be further developed in this thesis.

1.2.2 Planetary magnetospheres

Figure 1.5: Schematic of the terrestrial magnetosphere showing the distended field lines of both day and night side magnetospheres. RM P denotes the distance to

sub-solar magnetopause. c Fran Bagenal & Steve Bartlett.

(43)

around the planet and, secondly, even if the central object is umagnetized or weakly magnetized its interaction with the magnetic field of the environment, say the solar wind, may create a magnetospheric-like system (like the planets Venus, Mars or Saturn’s moon Titan, and Jupiter’s moon Io, or even the interaction of a comet with the solar wind). On a much larger scale the entire heliosphere can be described as a magnetospheric-like system produced by the interaction of the sun (the central object) and the interstellar medium (the external medium). A basic configuration of a planetary magnetosphere is presented in Figure 1.5.

The ionized solar wind, is diverted around the magnetosphere. It actually hits the strongly magnetized planets magnetosphere and since it is super-Alfvénic and super sonic a bow-shock is formed (analogous to the sonic boom in supersonic aerody-namic flow past an obstacle). As a consequence, behind the bow shock, the shocked solar wind is slowed down, compressed and heated. The bounding surface of the magnetosphere is called the magnetospause. The magnetopause is defined as the discontinuity of the magnetic field, the region where the direction of the magnetic field changes: inside the magnetopause the controlling magnetic field is that of the planet, while outside it is the solar wind magnetic field. The region between the bow-shock and the magnetopause is called the magnetosheath, a term first introduced by Dessler & Fejer [1963]. Whatever are the details of the interaction between the solar wind and the magnetospheres, in nearly all cases, the interaction region has a magnetotail that can extend for long distance (several Rp ≡ planet radius) in the

night side of the magnetosphere.

(44)

1.3. Motivations and outline. 11 The curved nature of the bow shock means that at a given position and time one can distinguish between two different types of geometries determined by the angle between the normal to the shock and the interplanetary magnetic fields ΘBn (see

Figure1.6). In a quasi-perpendicular shock, (ΘBn∼ 90◦), usually referred as "quiet

magnetosheath" the transition in the plasma properties upstream and downstream the shock is abrupt. Whereas in a quasi parallel shock (ΘBn ∼ 0◦ or 180◦), which

is referred to as "disturbed magnetosheath" the transition is "fuzzier" (or broader) and characterized by large-amplitude magnetic field, velocity and density fluctua-tions. Noting that these boundaries are dynamically unstable (due essentially to the change in the dynamical pressure of the solar wind) and were shown to con-trol some properties of the turbulence within the magnetosheath [Sahraoui et al.,

2006; Yordanova et al.,2008]. They are also the cause that generates various wave phenomena, instabilities and large scale inhomogeneities (e.g., KelvHelmotz in-stability [Hasegawa et al.,2004]).

One of the reasons that makes the magnetosheath important in magnetospheric physics is that it is the interface of the solar wind-magnetosphere interaction, af-fecting the physical processes occurring within the magnetopause. It is in fact the magnetosheath magnetic field and plasma that interact with the magnetopause and, consequently with the magnetosphere. Therefore, studying the properties of turbu-lence in the magnetosheath should help to better understand the dynamical coupling between the solar wind and the magnetosphere by improving the current reconnec-tion models at the magnetopause (e.g.,Belmont & Rezeau [2001]). Indeed, most of those models consider a large scale "laminar" current sheet in which magnetic recon-nection proceeds to allow the solar wind particules to penetrate into the magneto-sphere [Lundin et al.,2003]. A more realistic model would be to consider the effects of turbulence in the upstream (magnetosheath) region whose properties remain to be determined. In addition to the relevant role of the magnetosheath turbulence to the problems of particles transfer through the magnetopause, its compressible nature (as we will see in the course of this thesis) makes it also relevant for a bet-ter understanding of other highly compressible astrophysical media that spacecraft cannot reach such as the interstellar medium or the supernova remnants [ Vázquez-Semadeni et al.,1996]. Furthermore, the fact that the magnetosheath is a bounded region, on one side by the bow shock and on the other side by the magnetopause, makes it useful to study the effect of large scales boundaries on the nature of the turbulence properties as we will see in Chapter 4.

1.3 Motivations and outline.

(45)

Cluster (magnetosheath of Earth) and THEMIS B/ARTEMIS P1 (in the solar wind at 1 AU). The comparative studies conducted during this thesis on the properties of turbulence in those different plasma environments could help developing (when possible) a unified description of the turbulence physics and phenomenology that is applicable to a variety of different systems.

The first motivation of this work was to explore the turbulence properties in the magnetosheath of Saturn from the MHD scales down to the electron ones using the set of plasma and wave instruments available for the first time in the Saturn’s environment. The waves instruments (gathered within the consortium of RPWS that will be described in Section 4.1.2) include in particular the tri-axial Search-Coil magnetometer (built by LPP), which measures the high frequency magnetic field fluctuations. Prior to the arrival of Cassini-Huygens mission to Saturn in 2004, there were no dedicated mission to orbit continuously around this planet. Indeed, Saturn’s magnetoshpere was visited only 3 times by spacecraft which were only rapid flybys: Pioneer 11 in 1979, Voyager 1 in 1980 and Voyager 2 in 1981. The properties of turbulence around Saturn remained therefore almost entirely unexplored. The interest in exploring the plasma turbulence near Saturn lies in its plasma con-ditions: it has more tenuous density and magnetic field, yielding a higher plasma β, Alfvén and turbulent Mach numbers that are not available for near-Earth space (see Table 1.2_{). The Mach numbers could indeed reach very high values (up to ∼ 100)}

comparable to the ones expected in highly compressible astrophysical media such as supernova remnants or accretion disks [Masters et al.,2013]. Making compara-tive studies between Earth and Saturn should allow us to study turbulence in wide range of plasmas conditions, which could serve as a proxy for studies of astrophysical turbulence that spacecraft cannot reach.

Distance (A.U.) 0.5 (Mercury) 1 (Earth) 10 (Saturn)

SW MS SW MS SW MS n (cm−3₎ ₆ ₅₀ ₄ ₁₂ _0.005 _0.5 B0 (nT) 6 70 5 20 0.5 2 Ti (eV) 18 600 50 250 5 300 ρi (km) 17 50 200 95 700 1800 βi 1 4 3 6 0.05 13 MA 4 1 6 2 15 7 Ms − − 0.05 0.2 − −

Table 1.2: Typical (approximated) plasma parameters at different radial dis-tances in the solar wind and in different planetary magnetosheaths (the density (n), background magnetic field (B0), ion Larmor radius (ρi = Vthi/2πfci where

fci= qB0/2πmi is the ion gyro-frequency), the ion plasma (βi), the Alfvénic Mach

(46)

1.3. Motivations and outline. 13 As we will see in the course of this thesis, the importance of compressible fluc-tuations in the planetary magnetosheaths in comparison with the solar wind raises questions about the applicability of the theoretical models based on incompressible MHD turbulence, which have been used to investigate large scale solar wind to plan-etary magnetosheaths turbulence and so underlies the need to develop more realistic models to explore those complex media. While a realistic model of turbulence that would include kinetic effects or plasma instabilities (both observed in the magne-tosheath, see e.g., Sahraoui et al. [2006]) remains out of reach, the new exact law for compressible isothermal MHD turbulence derived by Banerjee & Galtier [2013] provides the most general theoretical framework available to analyze the effect of compressibility in turbulent plasmas. I first used the model in a weakly compress-ible medium, namely the (fast and slow) solar wind, which has been extensively studied in the past three decades using the incompressible MHD model ofPolitano & Pouquet [1998], before applying it to a more compressible (and more complex) medium that is the magnetosheath of Earth.

Throughout my thesis, using in-situ spacecraft data from the Cassini, Cluster and THEMIS B/ARTEMIS P1 satellites, I tried to answer the following questions or-dered in three separate categories:

• What are the scaling laws of the magnetosheath turbulence at MHD and kinetic scales ? Are they "universal" or do they depend on the local (or distant) plasma conditions ?

• What is the nature of the plasma fluctuations (Alfvénic ? magnetosonic ?) that carry the energy cascade from the MHD to the sub-ion scales in the magnetosheath ? How do they compare with the solar wind observations ? • What is the role of the compressible fluctuations in the solar wind and the

magnetosheath ? How do they affect the cascade (dissipation) rate ? Do they influence the spatial anisotropy of the turbulence ? How do they depend on the turbulent Mach number ? Is the Iroshnikov-Kraichnan phenomenology applicable in compressible MHD turbulence or a new phenomenology is needed to characterize the energy cascade ?

I tried to write my thesis from a pedagogical point of view so to make it accessible and useful to advanced researchers who want to know some specific details regarding these three points, but also for students and new researchers in the field.

(47)

Chapter 3 gives a brief overview of the main observational, theoretical and numer-ical works of turbulence done in the solar wind and the magnetosheaths of Earth and Saturn (scaling laws, wave modes identification, mono/multifractlity, and com-pressibility).

Part II focuses on my research work on compressible turbulence in the planetary magnetosheaths and the solar wind. It divides into three chapters as well: Chapter4

is dedicated to the investigation of the magnetosheath of Saturn using Cassini data, and so the properties of turbulence are studied with a comparison to the mostly known ones in the solar wind. Those results are accompanied with a more detailed analysis performed in the magnetosheath of Earth using Cluster data. Data selec-tion and caveats are addressed as well with a brief overview regarding the different plasma and the waves instruments I used from Cassini spacecraft. Chapter 5, using the exact compressible model of isothermal MHD turbulence, I present the effect and the role of the compressibility in the fast and the slow wind separately using the THEMIS B/ARTEMIS P1 spacecraft data. Eventually, in Chapter 6, I show the first application of this model to a more compressible medium, the magnetosheath of Earth, using Cluster data, with a preliminary study regarding the scaling properties of turbulence.

(48)

Chapter 2

Theoretical Background

Contents

2.1 Fully developed turbulence. . . 15

2.2 Structure functions and intermittency . . . 17

2.3 Exact laws and phenomenologies . . . 19

2.3.1 Exact law for incompressible HD turbulence and K41 phe-nomenology . . . 19

2.3.2 Exact law for incompressible MHD turbulence and IK phe-nomenology . . . 21

2.3.3 Exact law for compressible isothermal MHD turbulence . . . 23

The goal of this second chapter is to present briefly the theoretical background on which my research work is based. I show how the theoretical works first done in incompressible hydrodynamics (hereafter HD) and MHD were refined taking into account compressibility.

The chapter consists of three main sections. Section 1 is focused on the fundamental concepts of hydrodynamical turbulence, "the less complicated" theory in turbulence we know. Section 2, points out the main statistical tools for studying turbulence. Section 3, deals with the three exact laws and the corresponding phenomenologies (when present) derived for incompressible HD [Kolmogorov, 1941], MHD [Politano & Pouquet,1998], and compressible MHD [Banerjee & Galtier,2013].

2.1 Fully developed turbulence

The beginning of the scientific studies of turbulence is marked with Osborne Reynolds observations of transition from a laminar to a turbulent flow in water pipe [Reynolds,

1883]. His observations led to the identification of a single dimensionless parameter, called the Reynolds number, and denoted:

Re =

U L

ν (2.1)

(49)

a

b

c

d

Figure 2.1: Transition from a (a) laminar to a (d) turbulent regime as function of the Reynolds number Re for a flow passing a cylindrical obstacle. a) Re = 1.54, b)

Re= 9.6, c) Re= 13.1, d) Re = 26. Adapted fromFrisch [1995].

On Figure 2.1 we see how the flow passes from laminar to a fully developed state when the Reynolds number Re increases to ≫ 1. We recall that Re expresses the

relative importance of inertial and viscous forces. A heuristic argument emerges by comparing the inertial and the dissipation terms of the Navier-Stokes equations (2.2) for incompressible fluids (ρ = constant):

∇ · u = 0 ∂u

∂t + (u · ∇) u = −∇P + ν∇

2_{(u) + F} (2.2)

(50)

2.2. Structure functions and intermittency 17 the viscous term, ∇P denotes the pressure gradient and F represents the external force injecting the energy. The nonlinear and the viscous terms have dimensions respectively [U_[L]2] and ν_[L[U ]2_]. The nonlinear term clearly dominates over the viscous

term when

Re =

[L][U ]

ν ≫ 1 (2.3)

In the Kolmogorov phenomenology, the previous argument defines the inertial range characterizing the turbulent cascade: it is characterized by the range of scales where dissipation is negligible in comparison with the nonlinear effect and also much smaller than energy containing scales. The energy is therefore transfered through scales with a cascade rate that is constant in the inertial range. This cascade rate equals the rate at which energy is injected at the largest scale into the system and the rate at which it is dissipated at the very small scales. The estimation of this cascade rate for compressible turbulence in the solar wind and the magnetosheath is part of my thesis work.

2.2 Structure functions and intermittency

A self-consistent statistical tool to characterize a random turbulent process is the probability density function (PDF) P (v) of a continuous random variable like the velocity v of a fluid element, which is defined as P(v)dt = P (t < v < t + dt). For a given turbulent process, the velocity fluctuations at different spatial scales ℓ can be approximated by the increments, defined as:

δvℓ(t) = vℓ(x + ℓ) − vℓ(x) (2.4)

As a consequence, the relative likelihood of the turbulent process at different scales ℓ can be investigated by estimating the probability density P(δvℓ). We will see in

Chapter 4 how the behaviour and the shape of the PDFs play an essential role in characterizing the multifractal character of the turbulence.

Multifractality is a phenomenon in which a system cannot be reproduced by a magnification of some part of it: zooming in and zooming out in the system reveals the irregular variation of the dynamics even if it is apparently periodic and chaotic. In other words, multifractality characterizes large sharp fluctuations in the turbulent fields. In order to describe statistically those bursty events, higher order moments of P(δvℓ) are needed.

The n-th moment of a probability distribution function is obtained by integrating the appropriate power of a random variable over all possible values. If the variable represents the increments at the different scales ℓ defined in Equation (2.4), the statistical moments of the different orders are called the structure functions:

Sn(ℓ) =< (δvℓ(x))n>=

Z +∞

−∞ P(δvℓ

(51)

The different moments give information about the shape of the probability density function, and vice versa. One can easily see that the zero-th order moment of the structure function is the total probability (i.e. one), the 1st order is the mean, for stationary process one can easily verify that the 2nd _{order structure function is}

related to the autocorrelation function defined in Equation (2.6).

R(ℓ) = hu (x)u (x + ℓ)i (2.6)

However, the structure functions are proved to be more constructive than the corre-lation functions in turbulence theory. One should note two main interests regarding the study of the structure functions. By varying the scale ℓ and keeping n constant we determine particular features of turbulence at different scales. Whereas varying n for a particular scale ℓ is suitable to study intermittency. Another interest of the structure functions arises from their scaling properties which is found to obey a scaling law within the inertial range [Kolmogorov,1991] :

Sn(ℓ) = anℓξn (2.7)

where an is a proportionality constant. For statistically monofractal processes, the

scaling exponents ξnare a linear function of the order n; deviations from this linear

behaviour corresponds to multifractality. The third order moment of the structure function (n = 3) defines one of the most famous laws of a fully developed turbulence, the Kolmogorov law (discussed in Section 2.3.1), which implies the linear relation ξn= n/3.

It is also useful to define the normalized versions of the 3rd _{and the 4}th _order

mo-ments, the skewness and kurtosis respectively because of their essential role in char-acterizing the shape of the PDFs. The skewness is defined as the third order moment divided by the three-halves power of the second order moment; i.e.,

S(ℓ) = S3(ℓ) (S2(ℓ))3/2

(2.8) A non-zero skewness indicates a skewed or asymmetric PDF, which means that larger excursions in one direction are more probable than in the other direction. The kurtosis K (or the flatness F = K − 3) is defined as the fourth order moment divided by the square of the second; i.e.,

K(ℓ) = S4(ℓ) (S2(ℓ))2

(52)

2.3. Exact laws and phenomenologies 19 to be estimated accurately in the sense that larger statistical samples are required. A review on a more detailed check on convergence tests could be found inDudok de Wit[2004].

Figure 2.2: A schematic showing the importance of the higher-order statistics. The higher is the moment, the more information it gives regarding the few bursty events in the tails of the PDFs.

We note that in turbulence studies, we use stationary (more precisely weakly sta-tionary) time series, in which the mean and the variance (and possibly higher order moments) do not change significantly when shifted with time. Moreover, it is neces-sary to use long time intervals with respect to the scales we are interested in order to apply the ergodic theorem which ensures that time averages coincide with ensemble averages.

2.3 Exact laws and phenomenologies

2.3.1 Exact law for incompressible HD turbulence and K41 phe-nomenology

The first exact law of a fully developed incompressible turbulence was derived by Kolmogorov in 1941 [Kolmogorov, 1941], known as the 4/5 law. Starting from the basic continuity and momentum conservation equations for an incompressible flow (Equation 2.2) one can derive analytically the Kolmogorov law using velocity correlation tensors between two points separated by the distance l.

Considering scales much larger than the dissipative ones ldissip and much smaller

than the injection ones l0, and using the classical homogeneity, stationarity, isotropy

assumptions of the turbulent fluctuations [Frisch,1995], we obtain:

−45εl =< (δvl)3> (2.10)

where δvl is the velocity increments at the scale l and ε is the energy cascade (or

(53)

it was created. This has implications of which the most important is that for small scales that belong to the inertial range, the turbulence exhibits universal behaviour. The second important implication is that the turbulence is locally isotropic, since any anisotropy would mean that the details of large scales still have an influence (note however that this do not apply to spatial anistropy due to the presence of a magnetic field in plasmas, which persists in the system even at the very small scales).

It is noteworthy that Equation (2.10) shows that the third-order moment of the ve-locity structure function scales linearly with the corresponding length scale l inside the inertial range. This relation can therefore be used to define the inertial range in the physical space.

Figure 2.3: Illustration of the Richardson-Kolmogorov cascade, in the real space (a) and Fourier space (b).

This exact law (unsigned third order law) could be rederived using dimensional analysis, a powerful tool in the arsenal of a student of turbulence. This phenomeno-logical approach is based on the turbulent image first suggested byRichardson[1922] who proposed the highly influential cartoon that characterizes the turbulent flows as composed by eddies of different size interacting with each other at the large scales and breaking up into smaller ones, and so on (Figure 2.3-(a)). The energy is trans-ferred and cascaded from the large scales of the motion to smaller scales forming the inertial range until it reaches a sufficiently small length scale such that the viscosity of the fluid can effectively dissipate the kinetic energy (Figure 2.3-(b)).

Let us restrict our analysis to the inertial range and consider eddies of size l. To these eddies we associate a velocity vl and a characteristic time required for a

com-plete distortion of the eddies τeddie (known as the eddie turnover time). In these

conditions one can define the average transfer rate of the (kinetic) energy from one scale to another as:

ε = εl≡ dEl dt ∼ vl2 τtr ∼ vl2 τeddie ∼ vl3 l ⇒ vl= εℓ 1/3 _(2.11)

(54)

iner-2.3. Exact laws and phenomenologies 21 tial range, the energy is supposed to be neither injected nor dissipated, the average energy transfer rate εl should be equal to the average energy dissipation or

injec-tion rate ε. Using the definiinjec-tion of the energy spectral density in homogeneous turbulence, we can obtain the following K41 phenomenological relation:

Ek∼ vl2k−1 ∼ (εk−1)2/3k−f 1∼ ε2/3k−5/3 (2.12)

This approximate relation can be converted into an equation by introducing a di-mensionless constant αk, called Kolmogorov’s constant, determined experimentaly

to be about ∼ 1.4 or ∼ 1.6. The equation is written as:

Ek= αkε2/3k−5/3 (2.13)

Recently, the incompressible Kolmogorov’s 4/5 exact law has been generalized to compressible fluid turbulence described within the isothermal and the polytropic closure equations [Galtier & Banerjee,2011;Banerjee & Galtier,2014].

2.3.2 Exact law for incompressible MHD turbulence and IK phe-nomenology

Similarly to the incompressible hydrodynamic turbulence, an exact relation for in-compressible MHD turbulence was derived in terms of the Elsässer variables [ Poli-tano & Pouquet, 1998]. These Elsässer variables, which were first proposed by

Elsasser [1950], combine the magnetic and the velocity fields and are particularly relevant to study incompressible MHD turbulence:

z± _{≡ v ± v}A (2.14)

where vA= b/√µ0ρ is the Alfvén speed. Writing the basic MHD equations in terms

of the Elsässer fields and assuming full isotropy, homogeneity, and incompressibility, and that the dissipation remains non zero as the Reynolds number goes to infinity, they obtained the exact relations corresponding to longitudinal structure functions:

−43ε±ℓ =< (δz±· δz±)δz ∓

ℓ > (2.15)

where ε± _{denotes the mean rate of pseudo-energies input flux (E}± ₌ 1

2z±· z±).